Full coverage auto insurance is different from what is called Public Liability and Property Protection, or PLPD.
PLPD covers only the damage you do to someone else's property or vehicle, or injuries you cause them when you are at fault in an accident.
It doesn't cover your own vehicle's damage.
Many people opt for PLPD for older cars that they won't get repaired if in a major accident, or if they are simply trying to save some money and would rather risk paying for damage themselves rather than paying that cost every month.
Full coverage auto insurance means that your car is covered in the case of an accident when it comes to damage that needs to be repaired. The reason that many finance companies demand that you keep full coverage is because that car is the collateral for your auto loan and if it's damaged but doesn't get repaired, their loan is at risk as well.
However, not all auto repairs are reimbursed under full coverage auto insurance. If your transmission goes out or you run over a nail and need a new tire, you need to pay for these things yourself.
It's only in the event of an accident or vandalism that your car repairs are covered. To really understand all the coverage your insurance offers, read your policy carefully or talk to your agent for details.
When you do talk to your agent be sure to ask him which policy and coverages give you the best protection. If you don't ask, your agent may think you know exactly what your looking for.